91Ó°ÊÓ

a. Suppose \(f(z)=\sum a_{n} z^{n}\) has radius of convergence 1 and assume that an analytic continuation of \(f\) has a pole at \(z=1 .\) Show that \(\sum a_{n} z^{n}\) diverges at every point on the unit circle. (Hint: Show that if \(\left\\{a_{n}\right\\} \rightarrow 0\), then \((1-z) f(z) \rightarrow 0\) as \(z \rightarrow 1\) from below, along the \(x\) -axis.) b. Generalize the result; i.e. show that if \(f(z)=\sum a_{n} z^{n}\) has a positive radius of convergence and an analytic continuation of \(\sum a_{n} z^{n}\) has a pole at any point on its circle of convergence, then \(\sum a_{n} z^{n}\) diverges at all points on the circle of convergence.